Concept explainers
An estimator θ is said to be consistent if for any ∈ > 0,
Now identify Y with
Show that
Explanation of Solution
Calculation:
Chebyshev’s inequality can be rewritten as:
The random variable considered here is the sample mean,
The quantity
Replace Y by
When
As a result, when
Thus, using Chebyshev’s inequality, it can be shown that
Want to see more full solutions like this?
Chapter 6 Solutions
Bundle: Probability and Statistics for Engineering and the Sciences, 9th + WebAssign Printed Access Card for Devore's Probability and Statistics for ... and the Sciences, 9th Edition, Single-Term
- Show that (X+1)/(n+2) is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?arrow_forwardShow that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.arrow_forward2) The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (0, 0+1), where is the true but unknown voltage of the circuit. Suppose that Y₁, ..., Yn denotes a random sample of such readings. a) Show that Y is a biased estimator of 0, and compute the bias. b) Find a function of Y that is an unbiased estimator of 0. 2) Sunnogo that the random vario obcorvation tributionarrow_forward
- 4. Suppose X1,... , Xn is a random sample of size n drawn from a Poisson pdf where A is an unknown parameter. Show X X is an unbiased estimator for Xarrow_forwardAssume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .arrow_forward29. Suppose X1, X2, X3 are a random sample of size 3 from a distribution with pdf f(x) = (1/0)ex/0 for x > 0, 0 > 0. Let ô₁ = X1, 02 = (X1+ X2)/2 and 3 = (X1+2X2)/3. Show that these estimators of 0 are all unbiased, and determine the relative efficiencies between them.arrow_forward
- Let Y₁, Y2, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0,0). Consider = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that is a biased estimator for 0.arrow_forwardSuppose X₁,..., Xn when n > 2 are i.i.d Bernoulli(p) where 0 < p < 1 is unknown. (a) Find a sufficient statistic T(X₁,..., Xn) for p. (b) Show that 1 {X₁ = 1, X₂ = 0} is an unbiased estimator of p(1-p), where 1 {} denotes the indicator function. (c) Use the Rao-Blackwell theorem to improve the above estimator.arrow_forwardConsider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θI got MLE = n/-∑logXi and UMVUE = (n-1)/∑logXi. Need help in proving consistencyarrow_forward
- Let Y1, Y2, ..., Yn be a random sample with E(Y;) = µ and V (Y;) = o². Show that (Yi - is a biased estimator for o² and that (Y; – Y)² is an unbiased estimator for o².arrow_forwardB) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.arrow_forwardEX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.arrow_forward
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
- Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman